An-Najah National University Faculty of Graduate Studies GIS-Based Hydrological Modeling of Semiarid Catchments (The Case of Faria Catchment) By Sameer ‘Mohammad Khairi’ Shhadi Abedel-Kareem Supervisors Dr. Hafez Q. Shaheen Dr. Anan F. Jayyousi Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in Water and Environmental Engineering, Faculty of Graduate Studies, at An-Najah National University, Nablus, Palestine 2005
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An-Najah National University Faculty of Graduate Studies
GIS-Based Hydrological Modeling of Semiarid Catchments
(The Case of Faria Catchment)
By
Sameer ‘Mohammad Khairi’ Shhadi Abedel-Kareem
Supervisors
Dr. Hafez Q. Shaheen Dr. Anan F. Jayyousi
Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science in Water and Environmental Engineering, Faculty of Graduate Studies, at An-Najah National University, Nablus, Palestine
2005
III
بسم الله الرحمن الرحيم
أنزل من السماء ماء فسالت أودية بقدرها فاحتمل السيل
)16(ألرعد .........زبدا رابيا
صدق االله العظيم
IV
… Dedicated to My parents, wife and my daughter (Muna)
V
Acknowledgments
First of all, praise is to Allah for making this thesis possible. I would like to
express my sincere gratitude to Dr. Hafez Shaheen and Dr. Anan Jayyousi
for their supervision, guidance and constructive advice. Special thanks go
also to the defense committee members.
Thanks also go to those who helped in providing the data used in this study.
Water and Environmental Studies Institute or giving me the chance to work
on GLOWA project, Palestinian Water Authority (PWA) for producing of
water data used in this study.
My parents, brothers and sisters, thank you for being a great source of
support and encouragement. All my friends and fellow graduate students,
thank you.
Special thanks to my friend Eng. Rami Qashou’ who’s support will never
be forgotten.
Finally, thanks to my dear wife Heba for her love, moral support and
patience.
VI
Table of Contents ACKNOWLEDGMENTS ........................................................................................................ V TABLE OF CONTENTS ........................................................................................................ VI LIST OF ABBREVIATIONS .......................................................................................... VIII LIST OF TABLES ....................................................................................................................... IX LIST OF FIGURES ...................................................................................................................... X ABSTRACT .................................................................................................................................... XII CHAPTER ONE INTRODUCTION .................................................................................... 1
1.1 BACKGROUND ............................................................................................................................ 2 1.2 OBJECTIVES ................................................................................................................................. 3 1.3 RESEARCH NEEDS AND MOTIVATIONS ....................................................................... 4 1.4 METHODOLOGY ........................................................................................................................ 7 1.5 DATA COLLECTION ................................................................................................................. 9
CHAPTER TWO LITERATURE REVIEW ............................................................... 11
2.1 HYDROLOGY OF SEMIARID REGIONS ........................................................................ 12 2.1.1 Climate and Rainfall ................................................................................................ 13 2.1.2 Runoff Generation and Channel Flow .......................................................... 16 2.1.3 Storages ............................................................................................................................ 17
2.2 RAINFALL-RUNOFF MODELING .................................................................................... 18 2.2.1 Historical Overview ................................................................................................. 19 2.2.2 Classification of Models and Basic Definitions ..................................... 21
2.3 USE OF GIS IN HYDROLOGY .......................................................................................... 23 2.4 PREVIOUS WORK IN THE STUDY AREA .................................................................... 26
CHAPTER THREE DESCRIPTION OF THE STUDY AREA ...................... 28
3.1 LOCATION AND TOPOGRAPHY ....................................................................................... 29 3.2 CLIMATE .................................................................................................................................... 33
3.2.1 Wind ................................................................................................................................... 33 3.2.2 Temperature ................................................................................................................... 34 3.2.3 Relative Humidity ..................................................................................................... 36 3.2.4 Rainfall ............................................................................................................................. 36 3.2.5 Evaporation .................................................................................................................... 38 3.2.6 Aridity of the Catchment ....................................................................................... 40
4.2.1 Density of Rain Gauges ......................................................................................... 60 4.2.2 Consistency of Rainfall Data .............................................................................. 62 4.2.3 Monthly Rainfall ........................................................................................................ 63 4.2.4 Annual Rainfall ........................................................................................................... 66 4.2.5 Trend Analysis ............................................................................................................. 68
4.3 EXTREME VALUE DISTRIBUTION ................................................................................. 71 4.3.1 Gumbel Distribution ................................................................................................ 72
CHAPTER FIVE RUNOFF MODELING ..................................................................... 79
5.1 INTRODUCTION ....................................................................................................................... 80 5.2 GIUH MODEL ......................................................................................................................... 81 5.3 TRAVEL TIME ESTIMATION OF THE KW-GIUH MODEL ............................... 83 5.4 STRUCTURE OF THE KW-GIUH MODEL ................................................................. 87 5.5 KW-GIUH MODEL INPUT PARAMETERS ............................................................... 91
5.9 ANALYSIS AND DISCUSSION ........................................................................................ 108 CHAPTER SIX CONCLUSIONS AND RECOMMENDATIONS .............. 111
REFERENCES ............................................................................................................................. 117 APPENDIX A (TABLES) ..................................................................................................... 125 APPENDIX B (FIGURES) .................................................................................................. 162 APPENDIX C (KW-GIUH OUTPUTS) .................................................................... 171 APPENDIX D (PICTURES) .............................................................................................. 179 ب ......................................................................................................................................................... الملخص
VIII
List of Abbreviations Symbol The meaning
KW-GIUH Kinematic Wave based Geomorphological Instantaneous Unit Hydrograph
GIS Geographical Information System DEM Digital Elevation Model EAB Eastern Aquifer Basin PHG Palestinian Hydrology Group PWA Palestinian Water Authority IDF Intensity Duration Frequency Curves MOT Meteorological Office of Transport WESI Water and Environmental Studies Institute N Optimal number of stations Ep Allowable percentage of error Cv Coefficient of variation
avP Mean of rainfall 1−nσ Standard deviation
P(x) Probability of exceedance oixT Time for the flow to reach equilibrium ioq ith-order overland flow discharge per unit width
Lq Lateral flow rate iosh ith-order water depth at equilibrium icsQ ith-order channel discharge at equilibrium
rkxT Travel time for the channel storage component
ckxT Travel time for the channel translation component iOAP Ratio of the ith-order overland area to the catchment area
A Total area of the catchment iN ith-order stream number icL ith-order stream length
on Overland flow roughness cn Channel flow roughness
Ai ith-order sub catchment contributing area S io ith-order overland slope S ic ith-order channel slope
ji xxP Stream network transitional probability ΩB Channel width at catchment outlet
Ω Stream network order
IX
List of Tables Table 1: Abstraction from Wells in the Faria Catchment ........................... 42 Table 2: Spring Groups and Spring Information within Faria catchment .. 43 Table 3: Surface Water Quality Parameters for Badan and Faria Streams . 49 Table 4: Major Soil Types and Characteristics in Faria Catchment ........... 50 Table 5: Total Land use Cover of the Faria Catchment .............................. 56 Table 6: Available Rainfall Stations within the Faria Catchment .............. 60 Table 7: Monthly Rainfall Totals of Nablus Station (mm) ......................... 65 Table 8: Statistical Measurements of the Annual Rainfall of the Six
Stations of Faria Catchment .......................................................... 67 Table 9: Trend Equations for the 5-years Moving Average of the Six
Stations .......................................................................................... 69 Table 10: t and
1 , 22
nt α− −
with 90% and 95% Confidence Intervals .............. 71
Table 11: Parameters of Gumbel Distribution for the Six Stations of the Faria Catchment .......................................................................... 74
Table 12: Areal Rainfall Using the Thiessen Polygon Method .................. 76 Table 13: Coordinates, Elevations, Rainfall Averages and Estimated
Averages for the Six Stations ...................................................... 79 Table 14: ji xxP For the Three Sub-catchments ............................................ 95 Table 15: KW-GIUH Input Parameters for Al-Badan Sub-catchment ....... 96 Table 16: KW-GIUH Input Parameters for Al-Faria Sub-catchment ......... 96 Table 17: KW-GIUH Input Parameters for Al-Malaqi Sub-catchment ...... 98
X
List of Figures Figure 1: A flow Chart Depicting the General Methodology Followed in
this Study ....................................................................................... 9 Figure 2: Arid Regions around the World (UNESCO, 1984) ..................... 12 Figure 3: Hydrologic Cycle with Global Annual Average Water Balance
(Chow et al., 1988) ...................................................................... 15 Figure 4: Classification of Hydrological Models (Lange, 1999) ................ 23 Figure 5: Location of the Faria Catchment within the West Bank ............. 30 Figure 6: Springs and Wells within the Faria Catchment ........................... 31 Figure 7: Topographic Map of the Faria Catchment .................................. 32 Figure 8: Mean Monthly Temperatures in Nablus and Al-Jiftlik .............. 35 Figure 9: Spatial Distribution of the Mean Annual Temperature in the Faria
Catchment .................................................................................... 35 Figure 10: Rainfall Stations and Rainfall Distribution within the Faria
in Nablus and Al-Jiftlik ........................................................... 39 Figure 12: Potential Annual Evapotranspiration Rates in the Faria
Catchment ................................................................................. 40 Figure 13: Average, Maximum and Minimum Monthly Discharge of Total
Springs within Al-Badan Sub-catchment.................................. 44 Figure 14: Average, Maximum and Minimum Monthly Discharge of Total
Springs within Al-Faria Sub-catchment .................................... 45 Figure 15: Soil Types of the Faria Catchment ............................................ 51 Figure 16: Geology Map of the Faria Catchment ....................................... 52 Figure 17: Part of the airphotos of Faria Catchment ................................... 53 Figure 18: The New Land use Map of the Faria Catchment ....................... 57 Figure 19: Double Mass Curve for the Stations of Faria Catchment .......... 63 Figure 20: Mean Monthly Rainfall of the Six Stations of the Faria
Catchment ................................................................................. 64 Figure 21: Monthly Average Rainfall of Nablus Station Plotted As
Averages of Five Years Intervals.............................................. 66 Figure 22: Yearly Rainfall of Nablus Station ............................................. 67 Figure 23: The 5-year Moving Average of Nablus, Beit Dajan and Al-Faria
Stations ...................................................................................... 69 Figure 24: The 5-year Moving Average of Tubas, Taluza and Tammun
Stations ...................................................................................... 69 Figure 25: Gumbel Plots of Annual Rainfall for Nablus Station ................ 74 Figure 26: Gumbel Plots of the Standardized Variable of the Six Stations of
Faria Catchment ........................................................................ 75 Figure 27: Thiessen Polygon Map for the Faria Catchment ....................... 77
XI
Figure 28: Runoff Structure for a second-Order Catchment (Lee and Chang, 2005) ......................................................................................... 84
Figure 29: Surface Flow Paths of A third-Order Catchment (Lee and Chang, 2005) ............................................................................. 90
Figure 30: Digital Elevations Model (DEM) for the Faria Catchment ....... 93 Figure 31: The Three Sub-catchments of the Faria Catchment .................. 94 Figure 32: The Stream Order Networks for the Three Sub-catchments of
the Faria Catchment .................................................................. 95 Figure 33: 1mm-GIUH for Al-Faria and Al-Badan Sub-catchments ......... 99 Figure 34: 1mm-GIUH for Al-Malaqi Sub-catchment ............................... 99 Figure 35: Variation of GIUH with Excess Rainfall ................................... 99 Figure 36: Sensitivity Analysis of Channel Width on GIUH ................... 100 Figure 37: Sensitivity Analysis of Overland Roughness Coefficient on
GIUH ....................................................................................... 101 Figure 38: Sensitivity Analysis of Channel Roughness Coefficient on
GIUH ....................................................................................... 102 Figure 39: Al-Badan Sub-catchment and its Drainage Network .............. 103 Figure 40: Rainfall Depth and Infiltration Capacity Curve of 14/2/2004
Event ........................................................................................ 105 Figure 41: Recorded and Estimated Direct Runoff Hydrograph for Al-
Badan Sub-catchment, Event of 14/2/2004 ............................. 106 Figure 42: Rainfall Depth and the Phi-Index of Event of 5/2/2005 .......... 107 Figure 43: Recorded and Estimated Direct Runoff Hydrograph for Al-
Badan Sub-catchment, Event of 5/2/2005 ............................... 108
XII
GIS-Based Hydrological Modeling of Semiarid Catchments
Figure 21: Monthly Average Rainfall of Nablus Station Plotted As Averages of Five Years Intervals
4.2.4 Annual Rainfall
Rainfall data for five stations were obtained from PWA and from Nablus
meteorological station for Nablus station. The rainfall data for the stations
for the 50 last years are presented in Appendix A6.
Statistical analysis has been utilized for the rainfall data of the six stations
of the Faria catchment. This includes the annual average (AVG), the
standard deviation (STD), the maximum (MAX) and the minimum (MIN)
rainfalls recorded by these stations as tabulated in Table 8.
67Table 8: Statistical Measurements of the Annual Rainfall of the Six Stations
of Faria Catchment
From the table it is noticed that Nablus and Taluza stations have the largest
average annual rainfalls, whereas Al-Faria station has the lowest average
annual rainfall. Tubas station has an average annual rainfall of 415 mm,
which nearly equals the arithmetic average of the annual average rainfalls
of the six stations of the Faria catchment at about 430 mm. In general,
rainfall decreases from north to south and west to east. All stations are
functioning except Al-Faria meteorological station which was put in still in
the year 1989.
The annual rainfall for Nablus station is as shown in Figure 22. The full
time series for all six stations are enclosed in Appendix B1.
0
500
1000
1500
46-4
7
50-5
1
54-5
5
58-5
9
62-6
3
66-6
7
70-7
1
74-7
5
78-7
9
82-8
3
86-8
7
90-9
1
94-9
5
98-9
9
02-
Years
Rai
nfal
l (m
m)
Figure 22: Yearly Rainfall of Nablus Station
Rainfall Station AVG(mm) STD MAX
(mm) MIN (mm)
Nablus Meteorological Station 642.6 203.3 1387.6 315.5 Taluza Primary School 630.5 196 1303.1 292.2 Tubas Secondary School 415.2 143.9 899.5 201.5 Beit Dajan Station 379.1 134.8 777 141 Tammon Primary School 322.3 106.4 616.1 124.2 Al-Faria Meteorological Station 198.6 83 424 30
68
From the figure it is noticed the highest total annual rainfall occurred in the
year of 1992 and reached about 1350 mm. In other words, the exceedance
probability for such extreme value is equal zero.
4.2.5 Trend Analysis
Simple trend analysis of the 5-years moving average is applied to the
available annual rainfall data of the six stations. The average of the first 5-
years rainfalls is calculated, and then the average of the next 5 records
excluding the first and including the sixth is computed. The procedure
continues by finding the average of the following 5 years and so on.
Figures 23 and 24 show the trends for the six stations, whereas Table 9
tabulates the results of the linear regression applied to the moving average
values of the stations.
0
200
400
600
800
1000
46-4
7
49-5
0
52-5
3
55-5
6
58-5
9
61-6
2
64-6
5
67-6
8
70-7
1
73-7
4
76-7
7
79-8
0
82-8
3
85-8
6
88-8
9
91-9
2
94-9
5
97-9
8
Time (years)
Rai
nfal
l (m
m)
Nablus Beit Dajan Al-Faria
Figure 23: The 5-year Moving Average of Nablus, Beit Dajan and Al-Faria Stations
69
0
200
400
600
800
1000
46-4
7
49-5
0
52-5
3
55-5
6
58-5
9
61-6
2
64-6
5
67-6
8
70-7
1
73-7
4
76-7
7
79-8
0
82-8
3
85-8
6
88-8
9
91-9
2
94-9
5
97-9
8
Time (years)
Rai
nfal
l (m
m)
Taluza Tubas Tammun
Figure 24: The 5-year Moving Average of Tubas, Taluza and Tammun Stations
Table 9: Trend Equations for the 5-years Moving Average of the Six Stations
Rainfall Station Trend Equation Y = (bX+a) r2
Nablus Meteorological Station 1.550 X + 601.5 0.1107 Taluza Primary School 0.565 X + 604.9 0.0172 Tubas Secondary School 0.594 X + 393.0 0.0514 Beit Dajan Station 1.327 X + 340.0 0.1352 Tammon Primary School 0.087 X + 313.5 0.0017 Al Faria Meteorological Station -1.363 X + 220.2 0.138
The table and the figures indicate that there is an increasing trend for all
stations except for Al-Faria station where the trend is decreasing. The
degree of trend which is reflected by the independent variable coefficient (b)
of the regression equation varies from 1.55 to 0.087 for the stations of
increasing trend.
70
The significance of the regression equations of the trends can be evaluated
by testing the hypothesis Ho: b=0. If this hypothesis is accepted, then
estimated Y equals mean Y, i.e.: ^Y Y
−
= , where ^Y is estimated Y and Y
−
is
mean Y, or the regression line does not explain a significant amount of the
variation in Y, where Y = bX + a.
In this situation one would use mean Y as an estimator for Y regardless of
the value of X. The hypothesis Ho: b=Bo=0 is equivalent to the hypothesis
Ho: r = 0, where r is the correlation coefficient. Test of hypothesis
concerning (b) can be done noting that
o
b
b Bs
⎛ ⎞−⎜ ⎟⎝ ⎠
5
has t-distribution with (n-2) degrees of freedom. Thus the hypothesis Ho:
b=Bo=0 versus Ho: b≠Bo is tested by computing
o
b
b Bts
⎛ ⎞−= ⎜ ⎟⎝ ⎠
6
Ho is rejected if
t > 1 , 2
2n
t α− −
The t-test has been conducted for 90% and 95% confidence intervals. The
above testing procedure is described in (Hann ,1977). The results of testing
are presented in Table 10. From the table it is noticed that the hypothesis
Ho: Bo=0 versus Ho: Bo≠0 is accepted for all stations except for Nablus and
Beit Dajan stations where the hypothesis is rejected. This means that there
is no significant trend of the 5-years moving average rainfall within the
Faria catchment except for Nablus and Beit Dajan stations.
71 Table 10: t and
1 , 22
nt α− −
with 90% and 95% Confidence Intervals
Rainfall Station t 1 , 22
nt α− −
95% 90% Nablus Meteorological Station 2.5 2 1.67 Taluza Primary School 0.95 2 1.67Tubas Secondary School 1.66 2 1.67 Beit Dajan Station 2.8 2 1.67 Tammon Primary School 0.29 2 1.67 Al Faria Meteorological Station 1.57 2.04 1.69
4.3 Extreme Value Distribution
The study of extreme hydrologic events involves the selection of a
sequence of the largest or smallest observations from the sets of data. For
example, the study of peak flows considers only the largest flow recorded
each year at a gauging station. Since these observations are located in the
extreme tail of the probability distribution of all observations from which
they are drawn (the parent population), it is not surprising that their
probability distribution is different from that of the parent population.
There are three asymptotic forms of the distributions of extreme values,
named Type I, Type II and Type III respectively. Type I which is known as
Gumbel distribution is used for its simplicity and publicity. It is the most
used distribution for rainfall and runoff data analysis especially for annual
records (Chow et al, 1988).
Hydrological systems are sometimes impacted by extreme events, such as
severe storms, floods, and droughts. The magnitude of an extreme event is
inversely related to its frequency of occurrence, very severe events
occurring less frequently than more moderate events. One way of analyzing
the rainfall data is using frequency analysis. The objective of the frequency
72
analysis of hydrologic data is to relate the magnitude of extreme events to
their frequency of occurrence through the use of probability distribution
(Chow et al., 1988).
4.3.1 Gumbel Distribution
For the purpose of extreme value analysis, the following procedure has
been followed:
1. The annual rainfall values were arranged in descending order over the
recorded period and each value was given a rank, r.
2. For each value of rainfall, denoted by x, the probability of exceedance,
P(x) was calculated using the Gringorten formula which is appropriate for
the analysis of extremes (Chow et al., 1988):
)12.0()44.0()(
+−
=nrxP 7
where:
x: annual rainfall
P(x): the probability of exceedance
r: the rank of x
n: the total number of recorded years
3. The probability of non-exceedance was calculated using the following
relation:
)12.0()44.0(
1)(1)(+−
−=−=nr
xPxF 8
73
where:
F(x): the probability of non-exceedance
x and P(x): as defined previously
4. The Gumbel probability distribution is defined as:
( )))(exp(exp)( gg axbxF −−−= 9
Where:
6σπ
=gb and, g
g ba γμ −= ( )5772.0=γ
μ : the mean of all rainfall values
σ : the standard deviation of the rainfall values
5. The estimated Gumbel value of rainfall is defined as follows:
( )( )( )[ ]g
g
ab
xFLnLnx +−−
=^
10
Annual rainfall values for the six stations in the Faria catchment were
substituted in the above equations of the Gumbel distribution and
parameters were calculated and listed in Table 11. The square values of the
correlation coefficient r2, are also determined. The Gumbel plots for one
station are presented in Figure 25. The Gumbel plots for all stations are
presented in Appendix B2. From the figures and tabulated results it is
confirmed that Gumbel distribution can be applied to model the annual
rainfall for the rainfall stations of the Faria catchment. From a statistical
point of view, all values of r2 are high, about 0.94, such that the suitability
of Gumbel distribution is assumed.
74 Table 11: Parameters of Gumbel Distribution for the Six Stations of the
Faria Catchment
0
200
400
600
800
1000
1200
1400
1600
0 0.2 0.4 0.6 0.8 1
Probability of Non-Exceedance, F(x)
Ann
ual R
ainf
all (
mm
)
Recorded Rainfall Estimated Rainfall
Figure 25: Gumbel Plots of Annual Rainfall for Nablus Station
To generalize things, the standardized variable ( )[ ]σμ−X is calculated for
the six stations. Gumbel distribution was applied to the standardized
variable and is plotted in Figure 26. From the figure it is clear that Gumbel
distribution is a good representative of the annual rainfall of the Faria
catchment.
Rainfall Station μ σ bg ag r2 Nablus Meteorological Station 642.6 203.3 0.0063 551 0.955
Taluza Primary School 630.5 196 0.0065 542.3 0.941Tubas Secondary School 415.2 143.9 0.0089 350.5 0.935 Beit Dajan Station 379.1 134.8 0.0095 318.4 0.95 Tammon Primary School 322.3 106.4 0.012 274.4 0.945 Al Faria Meteorological Station 198.6 83 0.0154 161.2 0.944
Standardized of all Stations 0 1 1.282 -0.45 1
75
-3
-2
-1
0
1
2
3
4
5
0 0.2 0.4 0.6 0.8 1
Probability of Non-Exceedance, F(x)
Ann
ual R
ainf
all (
mm
)
Recorded Rainfall Estimated Rainfall
Figure 26: Gumbel Plots of the Standardized Variable of the Six Stations of Faria Catchment
4.4 Areal Rainfall
A rain gauge records rainfall at a geographical point. In most of the
hydrologic analysis, average depth of precipitation over the area under
consideration ought to be computed. To calculate the spatially distributed
rainfall for an area, the point rainfall needs to be converted to areal rainfall.
There are several methods available in literature to estimate areal rainfall.
However, depending on the accuracy and the objective of the analysis, any
of the following methods can be used: arithmetic average, Thiessen
polygon, isohyetal, grid point, orthographic or isopercental method (Patra,
2001). In this study Thiessen polygon method has been used as explained
in the following section.
76
4.4.1 Thiessen Polygon Method
The first step in the Thiessen polygon method is to connect all the rain
gauges by straight lines so that no lines form an angle greater than 90
degrees. Next, perpendicular bisectors are constructed on the first lines.
The bisectors should intersect within the triangular areas. The area of each
polygon within the catchment is divided by the total area and expressed as
a percentage. The area percent multiplied by the rainfall amount for each
polygon gives an estimation of the rainfall over the catchment.
The relative area weight for six polygons enclosing the corresponding
stations of Faria catchment in percentages is presented in Table 12. Figure
27 shows the constructed Thiessen polygons in the study area. Areal Rain
Extension that works under ArcView GIS environment has been used to
delineate the Thiessen polygons and to calculate the areal rainfall.
Table 12: Areal Rainfall Using the Thiessen Polygon Method
Rainfall Station Polygon Area (km2) Weight% Rainfall
(mm)
Weighted Rainfall
(mm) Nablus Meteorological Station 36.2 0.11 642.6 69.6
Taluza Primary School 52.2 0.16 630.5 98.4 Tubas Secondary School 19.0 0.06 415.2 23.7 Beit Dajan Station 75.4 0.23 379.1 85.5 Tammon Primary School 48.5 0.15 322.3 46.8 Al Faria Meteorological Station 102.9 0.31 198.6 61.1
Total 334.3 1.00 385.2
The results indicate that the areal average rainfall for the whole Faria
catchment extending from Nablus Mountains to the Jordan River is 385
mm; which nearly equals the long term average rainfall of Beit Dajan
station.
77
#Y
#Y
#Y
#Y
#Y
#Y
Tubas
Nablus
Tammun
Al-Faria
Talluza
Beit Dajan
N
Prepared byEng. Sameer Shadeed
Thiessen Polygons Tubas Staion Talluza Station Nablus Station Al Faria Station Tammun StationBeit Dajan Station
#Y Rainfall StationsCatchment Boundary
0 3 6 9 Kilometers
Figure 27: Thiessen Polygon Map for the Faria Catchment
4.5 Correlation Analysis between Stations
The spatiality of rainfall distribution within the Faria catchment is
investigated by using the multiple regression analysis. Table 13 gives the
average annual rainfall, the x, y coordinates and the elevation of the six
stations. A relation between the average annual rainfall and the coordinates
and elevations of the rainfall are developed for five of the stations. The
sixth station, Taluza, was left to be used in verifying the results of the
regression analysis and to test the resulted regression equation. The
resulting equation due to regression analysis of the five stations is:
78R = 8308 _ 39.5X _ 2.6Y _ 0.3H 11
where:-
R: is the annual average rainfall in mm
X: is the x-coordinate in km
Y: is the y-coordinate in km
H: is the elevation in m
The square value of the correlation coefficient r2, was determined at about
0.985 which indicates a high correlation between rainfall, coordinates and
elevations. The developed relation was used to calculate the annual average
rainfall for the stations and the outcome are tabulated as in Table 13.
Table 13: Coordinates, Elevations, Rainfall Averages and Estimated Averages for the Six Stations
From the table it is clear that the estimated average annual rainfall is close
to the recorded one for Taluza station which is a verification of the results
with an error of about 2%. It is then to assume the suitability of the
developed relation.
Station Name X
Coord. (km)
Y Coord.(km)
Elev.(m)
AVG (mm)
Estimated AVG (mm)
Nablus Meteorological Station 178 178 570 642.6 643.2
Taluza Primary School 178 186 500 630.5 642.6 Tubas Secondary School 185 192 375 415.2 388.8 Beit Dajan Station 185 178 520 379.1 370.4 Tammon Primary School 187 188 340 322.3 351.0 Al Faria Meteorological Station 196 172 -237 198.6 189.9
79
CHAPTER FIVE
RUNOFF MODELING
80
5.1 Introduction
The computation of flow hydrograph is of great importance to water
resources engineers and scientists. Among the most basic challenges of
hydrology are the quantitative understanding of the processes of runoff
generation and prediction of the flow hydrographs and their transmission to
the outlet. The high spatial variability in rainfall intensities and amounts
combined with variability in soil properties makes prediction of runoff
generation very difficult especially for ungauged catchments. Even in cases
where catchments are gauged, the period of record is often too short to
allow accurate estimates of the different hydraulic parameters.
Traditional techniques have been widely applied for the estimation of
runoff hydrographs at the outlets of gauged catchments using historical
rainfall runoff data and unit hydrographs derived from them. Such
procedures are questioned for their reliability due to the climatic and
physical changes in the catchment and their application to ungauged, arid
and semiarid catchments.
In the unit hydrograph theory it is assumed that the potential abstractions
are fully met before runoff occurs. This assumption is applicable to humid
regions, but is doubtful in arid and semiarid regions. For arid and semiarid
regions, the infiltration portion is higher than for humid regions due to
higher infiltration rates and dry soil antecedent moisture condition. The
infiltrability of the soil is high and the infiltration process will continue
significantly during the rainfall event. The amount of actual infiltration
may not satisfy the infiltrability of the soil. The evaporation losses are also
high in arid and semiarid regions and the evaporation process may occur
during the storm. Therefore the applicability of the unit hydrograph
81
approach in semiarid regions should be investigated as to its basic
assumption of satisfying the abstraction and neglecting the surface and
subsurface flow interaction during the rainfall-runoff process (Shaheen,
2001).
In the West Bank, which is characterized as semiarid region, hydrological
modeling has not been given enough care and no intensive studies have
been done. However, this study is an attempt to hydrologically investigate
and to derive the unit hydrograph for the Faria catchment as one of the
most important catchments of the West Bank, since the catchment has not
been modeled using appropriate rainfall-runoff model so far. The KW-
GIUH model that was developed for ungauged catchments is to be used in
the modeling.
5.2 GIUH Model
Hydrological simulation models can take the form of theoretical linkage
between the geomorphology and hydrology. The geomorphological
instantaneous unit hydrograph (GIUH) is one approach of these kinds of
models. The GIUH focuses on finding the catchment response given its
geomorphological features. The model uses catchment characteristics to
predict flow rates. GIUH can be applied to any excess rainfall through
convolution to produce the direct runoff hydrograph.
GIUH approach has been applied by several engineers to predict runoff
from rainfall for ungauged catchments. They have been proposed to
estimate floods for ungauged streams by using the information obtainable
from topographic maps or remote sensing possibly linked with the
82
Geographic Information Systems (GIS) and Digital Elevation Models
(DEM) (Snell and Sivapalan, 1994; Jain et al., 2000; and Hall et al., 2001).
Lee and Chang (2005) reviewed the development of GIUH approach and
concluded that the significant advance in research on the topographic
runoff approaches was the development of the geomorphologic
instantaneous unit hydrograph model (GIUH) proposed by Rodriguez-
Iturbe and Valdes (1979). During the last two decades, the use of
catchment geomorphologic characteristics in runoff simulations has
received a great deal of attention from hydrologists (e.g. Gupta et al., 1980;
Rodriguez-Iturbe et al., 1982; Kirshen and Bras, 1983; Karlinger and
Troutman, 1985; Agnese et al., 1988; Chutha and Dooge, 1990; Lee and
Yen, 1997; Yen and Lee, 1997; Olivera and Maidment, 1999; Berod et al.,
1999; Brooks and McDonnell, 2000).
The concept of the Geomorphological unit hydrograph is introduced by
Rodriguez-Iturbe and Valdes in 1979. This method is based on the Horton-
Strahler ordering law.
In the Strahler system for stream ordering and catchment ranking a stream
segment with no tributaries is called a first order segment. When two
stream segments of the same order meet, the order of the downstream
stream segment is raised with one. When two stream segments of a
different order meet, the order of the downstream segment is equal to the
highest order upstream.
In the GIUH approach, Rainfall excess is assumed to follow different paths
on overland areas and in channels of different stream orders
probabilistically, according to the drainage pattern to reach the catchment
outlet. The travel time of the rainfall excess is assumed to follow a
83
probability distribution in a channel of a given order. The exponential and
uniform distributions have been proposed by Gupta et al. (1980). Jin
(1992) suggested gamma distribution to get better results. Various methods
have been used to determine the time scale to be used with the probability
distribution. Rodriguez-Iturbe and Valdes (1979) gave the time parameters
as regression equations from discharge records. Agnese et al. (1988)
obtained the time scale formula from experimental data. Rodriguez-Iturbe
et al. (1982) estimated the first order channel travel time by using a
kinematic wave approximation. The travel times of higher order channels
were then related to the first order channels through geomorphologic laws.
An alternative approach was provided by Lee and Yen (1997). The travel
times for different orders of overland areas and channels were derived
using the kinematic wave theory and then substituted into the GIUH model
to develop a kinematic wave based GIUH model (KW-GIUH) for
catchment runoff simulation. The available KW-GIUH program (version
1.2) can be applied to catchments with stream network of up to the seventh
order. The program has been developed by Kwan Tun Lee and Chin-Hisn
Chang, Watershed Hydrology and Hydraulics Laboratory, Department of
River and Harbor Engineering and National Taiwan Ocean University. The
KW-GIUH model is applied in this study to predict the runoff hydrographs
of the Faria catchment.
5.3 Travel Time Estimation of the KW-GIUH Model
The travel time of the surface flow on a hillslope depends on the slope,
surface roughness, and flow depth. The travel times for different orders of
overland areas and channels were derived using the kinematic-wave theory
and then substituted into the GIUH model to develop a kinematic-wave
84
based GIUH model (KW-GIUH) for watershed runoff simulation. The
following is a discussion of the approach applied.
An ith-order sub-basin of the catchment is conceptually simplified as
consisting of two identical rectangular form V-shape overland flow planes.
Each plane contributes a lateral discharge into a channel of constant cross
section and slope as shown in Figure 28.
The mean length of the ith-order V-shape overland flow plane is
c
ii
ci
OAo
LNAP
L2
= 12
Where A is the total area of the catchment, iOAP is the ratio of ith-order
overland area to the total catchment area, iN is the number of the ith-order
channels and icL is the mean channel length of the ith-order sub-basin.
Figure 28: Runoff Structure for a second-Order Catchment (Lee and Chang, 2005)
The flow rate at the end of a plane increase with time until it reaches
equilibrium. The longest time for a raindrop to travel through the ith-order
85
overland plane oixT is the time for the flow to reach equilibrium in the plane.
Thus, the discharge for the ith-order overland sub-basin at any time, oixTt < ,
is
mL
o
oo tq
nSq i
i)(
21
= 13
Where ioq is the ith-order overland flow discharge per unit width, ioS is the
mean ith-order overland slope, on is the overland flow roughness
coefficient, m is a constant and Lq is the lateral flow rate joining the main
flowioq .
Once the flow equilibrium state reached, the discharge afterward, oixTt > , is
mos
o
ooLoso i
iii
hn
SLqqq21
1=== 14
Where iosq and
iosh represent the ith-order overland discharge and water
depth at equilibrium, respectively. The travel time for the ith-order
overland plane is m
mLo
oo
L
osx
qS
Lnqh
Ti
ii
oi
1
121 ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛==
− 15
Since the sub-basin model is conceptually composed of two identical
rectangular planes, the lateral discharge into the central channel is
contributed from both side planes. Thus, the lateral discharge
becomes ioL Lq2 . The amount of rain that falls directly onto the channels is
small compared to that falling on overland planes and can therefore be
neglected. As the flow in the ith-order overland planes gradually increases with time to reach equilibrium, an additional time
ixT is needed for the ith-
order channel to reach its equilibrium. A first order channel conveys only
86
the lateral discharge contributed by two first order overland slopes. Thus, the discharge of the first order channel at any
1xTt < is
moL
c
cc t
BLq
nSBQ ⎟
⎟⎠
⎞⎜⎜⎝
⎛=
1
21
1 11
1
2 16
Where 1B is the first order channel width, 1cS is the mean slope of the first
order channel and cn is the roughness coefficient of the channel flow. And
for 1xTt ≥
mcs
c
ccoLcsc h
nSBLLqQQ
1
11111
21
12 === 17
Where 1csQ and
1csh represent the first order channel discharge and water
depth at equilibrium, respectively. Therefore, the rainwater travel time for
the first order channel is m
c
cocL
oLoL
csx
SB
LLnqLq
BLq
hBT
1
21
1
11
1
11
11
1
1
222 ⎟
⎟
⎠
⎞
⎜⎜
⎝
⎛== 18
Since the catchment is considered as a multiple sub-basin system, the water
is transported successively from lower order to higher order channels. Thus, the discharge for an ith-order (i>1) channel at any time
ixTt < is
m
i
oLco
c
cic t
BLqh
nSBQ ii
⎟⎟⎠
⎞⎜⎜⎝
⎛+=
211
21
19
Where icoh is the water depth at the entrance of the ith-order channel. By
considering that the rainwater travels from upstream sub-basins through different paths towards the ith-order sub-basins,
icoh can be expressed as
( ) m
cii
OAiicLco
i
i
i SBN
APANnqh
1
21⎥⎥⎦
⎤
⎢⎢⎣
⎡ −= 20
87
Likewise, for ixTt ≥ , the channel discharge at equilibrium is
mcs
c
cicoiL
mco
c
cicsici i
iii
i hnSBLLqh
nSBQQ
2121
12 =+== 21
Where icsQ and
icsh represent the ith-order channel discharge and water depth
at equilibrium, respectively. Hence, by replacing the 1csh with
1cocs hhi− (which represents the increase in flow resulting from lateral flow)
and deriving icsh from (10), the travel time for the ith-order channel become
( )⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−⎟
⎟
⎠
⎞
⎜⎜
⎝
⎛+=−=
i
i
ii
ii
iii
i co
m
ci
cocLmco
oL
icocs
oL
ix h
SBLLnqh
LqBhh
LqBT
1
21
222
22
From the equations presented, the travel times for different order sub-
basins can be estimated analytically from overland and channel hydraulics
instead of relaying on the catchment specified empirical formulas.
5.4 Structure of the KW-GIUH Model
Overland flow over a permeable soil surface can occur when the rainfall
rate is greater than the infiltration capacity or when surface saturation exists
in regions near the stream (Lee and Chang, 2005).
When a unit depth of rain excess falls uniformly and instantaneously onto a
catchment, the unit rainfall excess is assumed to consist of a large number
of independent, noninteraction raindrops. Thus, the whole rainfall-runoff
process can be represented by tracing the rainfall excess moving along
different paths towards the catchment outlet to produce the outflow
hydrograph (Lee and Yen, 1997).
Based on the Strahler ordering scheme, a catchment of order Ω can be
divided into different states. For example, Figure 29 shows the possible
88
travel paths of the rain drops for a third-order catchment. Most of the
surface flow occurs on the low portions of the catchment (the shaded area
in Figure 29); after that, it goes into the adjacent channel and then flows
through the stream network to the outlet. Each raindrop falling on the
overland region will move successively from lower to higher order
channels until it reaches the outlet. The catchment geomorphology is
represented probabilistically based on the stream order, instead of
simulating the overland surfaces and channels by their individually actual
geometry as in the deterministic modeling. The ith-order overland regions
is denoted by xoi and xi represents the ith-order channel, in which i = 1, 2,…, Ω. If w denotes a specified runoff path Ω→→→→ xxxx jioi ..... ,the
probability of a drop of rainfall excess adopting this path can be expressed
as
( ) . ... ...i oi i i j kOA x x x x x xP w P P P P
Ω= ,where
ioi xxP is the transitional probability of the
raindrop moving from the ith-order overland region to the ith-order channel and ji xxP is the transitional probability of the raindrop moving from an ith-
order channel to a jth-order channel and is computed as
i
jixx N
NP
ji
,= 23
Where ijN is the number of ith-order channels contributing to the jth-order
channels and iOAP is the ratio of ith-order overland area to the total
catchment and is computed as
⎟⎠
⎞⎜⎝
⎛−= ∑
−
=jii xx
i
iiiiOA PANAN
AP
1
1
1 24
Where iA is the mean of the drainage area of order i. and is estimated as
89
∑=
=iN
jji
i
i AN
A1
1 25
It should be noted that Aji denotes not only the areas of the overland flow
regions that drains directly into the jth channel of order i, but it also
includes overland areas draining into the lower order channels tributary to
this jth channel of order i.
The travel time for the overland flow region and for the storage component
of a channel are assumed to follow an exponential distribution, but the
translation component of a channel is assumed to follow a uniform distribution. For the state kx , the travel time for the channel storage
component and channel translation component are rkxT and
ckxT , respectively,
and the total travel time is ckrkk xxx TTT +=
The IUH can be represented by the convolution of two groups of the
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125Appendix A (Tables)
Table A1 1: Monthly Climatic Average Data of Al-Farai Station ................. 127 Table A1 2: Monthly Climatic Average Data of Nablus Station. ................. 128 Table A2 1: CROPWAT 4 Output Results for Nablus Station ...................... 130 Table A2 2: CROPWAT 4 Output Results for Al-Faria Station .................... 130 Table A3 1: Basic Information of Wells Located within the Faria Catchment ..... 132 Table A4 1: Monthly Flow Discharge of Shibli Spring ................................. 137 Table A4 2: Monthly Flow Discharge of Faria Spring .................................. 138 Table A4 3: Monthly Flow Discharge of Miska Spring ................................. 139 Table A4 4: Monthly Flow Discharge of Dafna Spring ................................. 140 Table A4 5: Monthly Flow Discharge of Balata Spring ................................ 141 Table A4 6: Monthly Flow Discharge of Asubian Spring ............................. 142 Table A4 7: Monthly Flow Discharge of Duleib Spring ................................ 143 Table A4 8: Monthly Flow Discharge of Qudeira Spring .............................. 144 Table A4 9: Monthly Flow Discharge of Hamad & Beida Spring ................. 145 Table A4 10: Monthly Flow Discharge of Al-Jiser Spring ............................ 146 Table A4 11: Monthly Flow Discharge of Sidreh Spring .............................. 147 Table A4 12: Monthly Flow Discharge of Tabban Spring ............................. 148 Table A4 13: Monthly Flow Discharge of Abu Saleh Spring ........................ 149 Table A5 1: Flow Records of Al-Badan and Al-Faria Flumes ....................... 151 Table A6 1: Annual Rainfall Data of the Faria Catchment Stations .............. 160
126
APPENDIX A1
MONTHLY CLIMATIC AVERAGES DATA
127
Table A1 1: Monthly Climatic Average Data of Al-Farai Station
Element Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Mean Max. Temp. (oC) 19.5 20.2 24.3 29.1 34.6 37.1 39.4 38.5 36.6 33.5 27.9 21.5
Mean Min. Temp. (oC) 9.3 9.2 12.1 14.4 19.0 21.1 22.7 24.2 22.9 20.2 16.8 11.9
Figure B1 1: Yearly Rainfall of Nablus Station ............................................. 164 Figure B1 2: Yearly Rainfall of Tubas Station ............................................... 164 Figure B1 3: Yearly Rainfall of Taluza Station ............................................. 165 Figure B1 4: Yearly Rainfall of Beit Dajan Station ....................................... 165 Figure B1 5: Yearly Rainfall of Tammun Station .......................................... 166 Figure B1 6: Yearly Rainfall of Al-Faria Station ........................................... 166 Figure B2 1: Gumbel Plots of Annual Rainfall of Nablus Station ................. 168 Figure B2 2: Gumbel Plots of Annual Rainfall of Taluza Station ................. 168 Figure B2 3: Gumbel Plots of Annual Rainfall of Tubas Station .................. 169 Figure B2 4: Gumbel Plots of Annual Rainfall of Beit Dajan Station ........... 169 Figure B2 5: Gumbel Plots of Annual Rainfall of Tammun Station .............. 170 Figure B2 6: Gumbel Plots of Annual Rainfall of Al-Faria Station .............. 170
163
APPENDIX B1
TIME SERIES PLOTS OF ANNUAL RAINFALL FOR THE STATIONS IN THE FARIA
CATCHMENT
164
Yearly Rainfall of Nablus Station
0200400600800
1000120014001600
46-4
7
50-5
1
54-5
5
58-5
9
62-6
3
66-6
7
70-7
1
74-7
5
78-7
9
82-8
3
86-8
7
90-9
1
94-9
5
98-9
9
02-
Years
Rai
nfal
l (m
m)
Figure B1 1: Yearly Rainfall of Nablus Station
Yearly Rainfall of Tubas Station
0
200
400
600
800
1000
46-4
7
50-5
1
54-5
5
58-5
9
62-6
3
66-6
7
70-7
1
74-7
5
78-7
9
82-8
3
86-8
7
90-9
1
94-9
5
98-9
9
02-
Years
Rai
nfal
l (m
m)
Figure B1 2: Yearly Rainfall of Tubas Station
165
Yearly Rainfall of Taluza Station
0200400600800
100012001400
46-4
7
50-5
1
54-5
5
58-5
9
62-6
3
66-6
7
70-7
1
74-7
5
78-7
9
82-8
3
86-8
7
90-9
1
94-9
5
98-9
9
02-
Years
Rai
nfal
l (m
m)
Figure B1 3: Yearly Rainfall of Taluza Station
Yearly Rainfall of Beit Dajan Station
0
200
400
600
800
1000
46-4
7
50-5
1
54-5
5
58-5
9
62-6
3
66-6
7
70-7
1
74-7
5
78-7
9
82-8
3
86-8
7
90-9
1
94-9
5
98-9
9
02-
Years
Rai
nfal
l (m
m)
Figure B1 4: Yearly Rainfall of Beit Dajan Station
166
Yearly Rainfall of Tammun Station
0100200300400500600700
46-4
7
50-5
1
54-5
5
58-5
9
62-6
3
66-6
7
70-7
1
74-7
5
78-7
9
82-8
3
86-8
7
90-9
1
94-9
5
98-9
9
02-
Years
Rai
nfal
l (m
m)
Figure B1 5: Yearly Rainfall of Tammun Station
Yearly Rainfall of Al-Faria Station
0
100
200
300
400
500
46-4
7
50-5
1
54-5
5
58-5
9
62-6
3
66-6
7
70-7
1
74-7
5
78-7
9
82-8
3
86-8
7
90-9
1
94-9
5
98-9
9
02-
Years
Rai
nfal
l (m
m)
Figure B1 6: Yearly Rainfall of Al-Faria Station
167
APPENDIX B2
GUMBEL PLOTS OF ANNUAL RAINFALL FOR THE STATIONS IN THE FARIA CATCHMENT
168
Gumbel Plots of Annual Rainfall for Nablus Station
0
200
400
600
800
1000
1200
1400
1600
0 0.2 0.4 0.6 0.8 1
Probability of Non-Exceedance, F(x)
Ann
ual R
ainf
all (
mm
)
Recorded Rainfall Estimated Rainfall
Figure B2 1: Gumbel Plots of Annual Rainfall of Nablus Station
Gumbel Plots of Annual Rainfall for Taluza Station
0
200
400
600
800
1000
1200
1400
0 0.2 0.4 0.6 0.8 1
Probability of Non-Exceedance, F(x)
Ann
ual R
ainf
all (
mm
)
Recorded Rainfall Estimated Rainfall
Figure B2 2: Gumbel Plots of Annual Rainfall of Taluza Station
169
Gumbel Plots of Annual Rainfall for Tubas Station
0
100200
300
400
500600
700
800900
1000
0 0.2 0.4 0.6 0.8 1
Probability of Non-Exceedance, F(x)
Ann
ual R
ainf
all (
mm
)
Recorded Rainfall Estimated Rainfall
Figure B2 3: Gumbel Plots of Annual Rainfall of Tubas Station
Gumbel Plots of Annual Rainfall for Beit Dajan Station
0
100
200
300
400
500
600
700
800
900
0 0.2 0.4 0.6 0.8 1
Probability of Non-Exceedance, F(x)
Ann
ual R
ainf
all (
mm
)
Recorded Rainfall Estimated Rainfall
Figure B2 4: Gumbel Plots of Annual Rainfall of Beit Dajan Station
170
Gumbel Plots of Annual Rainfall for Tammun Station
0
100
200
300
400
500
600
700
0 0.2 0.4 0.6 0.8 1
Probability of Non-Exceedance, F(x)
Ann
ual R
ainf
all (
mm
)
Recorded Rainfall Estimated Rainfall
Figure B2 5: Gumbel Plots of Annual Rainfall of Tammun Station
Gumbel Plots of Annual Rainfall for Al-Faria Station
0
100
200
300
400
500
0 0.2 0.4 0.6 0.8 1Probability of Non-Exceedance, F(x)
Ann
ual R
ainf
all (
mm
)
Recorded Rainfall Estimated Rainfall
Figure B2 6: Gumbel Plots of Annual Rainfall of Al-Faria Station
171APPENDIX C (KW-GIUH Outputs)
Output C 1: KW-GIUH Results of 1-mm Excess Rainfall for Al-Badan Sub-catchment ...................................................................... 172
Output C 2: KW-GIUH Results of 1-mm Excess Rainfall for Al-Faria Sub-catchment.............................................................................. 174
Output C 3: KW-GIUH Results of 1-mm Excess Rainfall for Al-Badan Sub-catchment ...................................................................... 176
172
Output C 1: KW-GIUH Results of 1-mm Excess Rainfall for Al-Badan Sub-catchment
Kinematic-Wave based Geomorphic Instantaneous Unit Hydrograph by Kwan Tun Lee and Chin-Hsin Chang Watershed Hydrology and Hydraulics Laboratory Department of River and Harbor Engineering National Taiwan Ocean University Version 1.2, February 2001, All rights reserved. Station : Badan Date : 16/7/2005 Area = 85.28 km*km Phi index = .00 mm/hr Based flow = .00 cms Watershed channel network order: 4 Overland flow roughness coefficient: .300 Channel flow roughness coefficient: .030 Order Ni Lci(km) Ai(km*km) Poai Soi Sci ----------------------------------------------------------------- 1 41 1.379 1.370 .660 .1700 .1400 2 6 3.202 10.120 .310 .0920 .0620 3 2 5.027 40.730 .019 .1400 .0510 4 1 3.172 85.280 .011 .1350 .0290 i -> j Pxixj --------------- 1 -> 2 .610 1 -> 3 .340 1 -> 4 .050 2 -> 3 1.000 2 -> 4 .000 3 -> 4 1.000 Rainfall-Runoff Simulation Results: Time Rainfall Q-recorded Q-simulated (hr) (mm/hr) (cms) (cms) --------------------------------------- 0 .00 .00 .00 1 1.0 .00 2.91